I have not yet read the other answers.
Before I say anything about whether she is rude
or whether you are a pushover,
I MUST say . . . .
her child doesn't want to go to practice or be on the team.
So . . . . separate from, and in addition to,
your differences with this woman and her expectations
and her requests to you,
it seems clear (to me) that this woman is pushing one or more
of her children into activities that they don't want to participate in.
Which is not to say I'm against parents encouraging their children
to try some activities . . . lots of situations, lots of reasons.
But, imo, with THIS mother, there's a whole bunch of other stuff
going on.
Without attempting to figure out what (ALL!) is going on with her,
I see her as overwhelmed, unable to manage her children,
her household, her responsibilities.
So she attempts to relieve some of her overwhelmedness
by asking you, and others, to take a child here or there,
or other helpful favors to help her get through the day.
I do NOT think you are a pushover.
I think you are a very nice lady and you have agreed,
more than once, to help this mother out.
I think if you start saying NO, there might be some stress
between the two of you adults, and/or between
some of her children and some of your children.
However, I believe YOU should choose if and when you offer
special events or privileges to children who are not part of your family.
Which you have done.
If you can, you might tell this mother you won't be saying yes
to any of her requests for (a while? the immediate future?
a few weeks?). Which does not mean you won't INVITE
one or another of her children to come along with your children
on some occasions. But only when it's at YOUR initiation,
not at hers.
Good luck.